**Engendering connectivity from incommensurable 5-fold and 6-fold conceptual frameworks (Part #1)**

Introduction

Fundamental cognitive patterns assumed to be characteristic of belief systems

Animation of interaction and interlocking between cognitive patterns

Interactive design of cognitive pattern animation

Visualizing alternative stories through manipulation of animation design options

Polyhedral catalysts of global imagination

Design, pattern language and geometry

Dialogue implications of design and geometry

Systems perspective on three-dimensional cognitive configuration and communication

Triangulation, connectivity and "stitching": enabling coherent global system dynamics

Resonance, fullerenes and the Middle East?

Peace as a meta-pattern of resonance: psychosocial, implicit and emergent?

Incommensurable cognitive patterns and their symbolism

Conclusion

References

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Produced on the occasion of worldwide media focus on the patterned movement of a truncated icosahedron in Kiev, whilst the United Nations demonstrates its long-standing inability to elicit relevant intelligence in more creative response to ongoing massacres in the Middle East.

This is an exploration of the hypothesis that unique belief systems depend for their coherence on distinctive patterns typically embodied in geometrical symbols in two dimensions. On the basis of that **assumption**, the case tentatively explored here is that of the "incommensurability" of the 5-fold Star of Islam and the 6-fold Star of David of Judaism -- both symbols appearing on flags of the nations having those distinct faiths.

Efforts at the reconciliation of the perspectives embodied in these faiths are seen here to be as questionable as that of endeavouring to fit together the pieces of a jigsaw puzzle on a flat surface -- when the pieces do not "match" or "fit" together. That approach may be compared to the practice of the impatient in seeking to "force" the pieces together -- even "banging" on them to ensure that they lock together. Even if "successful", the resulting picture is "wrong". This metaphor has both conceptual implications and territorial implications. To employ a chemical metaphor, in the case of Judaism the pieces might be understood as 6-valent, in comparison with "5-valent" pieces of Islam. The guiding assumptions for this exploration are the much-quoted statements of Albert Einstein:

- The significant problems we face can not be solved at the same level of thinking we were at when we created them.
- To repeat the same thing over and over again, and yet to expect a different result, this is a form of insanity.

The approach taken here explores the possibility that the "pieces" only fit together on a three-dimensional surface, namely a sphere. It is the spherical geometry that then merits consideration, together with the challenge of how to get from any "mis-fitting" two-dimensional layout to a three-dimensional form. Of course, two-dimensional layouts are far more readily comprehensible. Hence the focus on them. However the three-dimensional layout has the potential of rendering comprehensible a far more elegant layout which may well exemplify intuitions characteristic of the faiths so dramatically opposed.

The approach follows from various earlier explorations of the potential of mathematics to offer a new perspectives on these issues, including: *Geometry of Thinking for Sustainable Global Governance* (2009), *Spherical Configuration of Categories -- to reflect systemic patterns of environmental checks and balances* (1994), *Using Disagreements for Superordinate Frame Configuration* (1993), and *Mathematical Theology: future science of confidence in belief* (2011).

In the case of the Middle East, the argument has previously been developed that the opposing cultures are widely recognized as highly competent in number theory and geometry (cf. *Wikipedia* *List of Jewish mathematicians*; Keith Devlin, *The Mathematical Legacy of Islam*, 2002; Keith Critchlow, *Islamic Patterns: an analytical and cosmological approach*, 1999; Issam El-Said, *Islamic Art and Architecture: the system of geometric design*, 2008; Center for South Asian and Middle Eastern Studies, *Islamic Mathematics*; 2000, G. Donald Allen, *Islamic Mathematics and Mathematicians*, 2000; *Wikipedia*, *Mathematics in Medieval Islam*).

Carra de Vaux continues to be quoted by Islamic scholars to the effect that : *they (the Muslims) were indisputably the founders of plane and spherical geometry, which did not, strictly speaking, exist among the Greeks* (*The Philosophers of Islam*, 1921).

**These skills do not however seem to have been brought to bear on the psychosocial dynamics which divide them in relation to territory -- other than with respect to ballistics**. Nor does there appear to have been any interest in suggesting that this possibility merits exploration, as previously discussed (*And When the Bombing Stops? Territorial conflict as a challenge to mathematicians*, 2000).

What follows is essentially the description of the development of an interactive visualization experiment to illustrate possibilities accessible through any web browser using scalable vector graphics. The focus is on the experimental design and modification of the visualization, effectively enabling imaginative exploration of possibilities -- both playfully and as offering a pattern language through which differences and their implications can be disputed in terms of design aesthetics. The experiment also highlights the potential of a visualization package, the appropriately named Stella Polyhedron Navigator, through which the "global" implications of such patterns can be explored, as previously suggested (*Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization*, 2008).

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